22b each scheme has about the same computational

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Unformatted text preview: the analyses of Section 2.1, we might expect that solutions obtained by (2.2.5) have a higher order of spatial accuracy than those obtained by either (2.2.2b) or (2.2.4) (cf. (2.1.4b, 2.1.6b, 2.1.7b). This would be enough to abandon schemes (2.2.2b) and (2.2.4), if it were the only di erence between the methods. Let's apply the methods to two simple examples. Example 2.2.1. Consider (2.2.1) with = 1 and a if 0 if ( )= x x x x> 0 0 : The solution of this problem is easily obtained by the method of characteristics (Section 1.3) as ( )= ( ; ) uxt x t which is a sloping ramp moving in the positive direction with unit speed. Let us, rather arbitrarily, choose = 1 10 and = 1 20 and solve this problem for several spatial locations and a few time levels by the forward time-forward space and forward time-backward space nite di erence schemes (2.2.2b) and (2.2.4), respectively. n The Courant number j = 1 2 in each case. The results are shown in Tables 2.2.1 and 2.2.2 and Figure 2.2.3. The solution obtained by the forward time-forward space sche...
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